Gas Dynamics

Kn = λ/L

λ = mean free path

L = characteristic length of a problem

For Kn << 1 the continuum assumption hold true and the navier stokes equations can be applied

(c= sqrt(gamma*R*T))

• Viscous effects, since the euler equations are the inviscid NS equations

• Heat Transfer

• Heat sources

• Gravity

Shocks

The derivative consists of a local part (change of local quantity with time) and a convective term. This convective term has the same velocity as the flow itself.

This is a lagrangian view, where the material derivative describes the change of a given property of a “fluid parcel“ moving with the flow.

Balance laws for momentum and total energy account for

• external forces

• energy inputs and losses

these can change the momentum and total energy

-> strictly speaking not conservation laws

(quasi) 1-D, inviscid, steady, adiabatic

No, only density/pressure

Ma = 1 only appears in the smallest cross section

No, shocks are non-isentropic, therefore they cannot.

The critical temperature is the static temperature at Ma = 1

Temperature relation results from the conservation of energy and doesnt rely on isentropic flows.

Pressure & density relations need the isentropic assumptions (they were developed through the temperature relation and then applying the p/rho^gamma = const for a constant entropy)

Ma increases:

• u increases

• p,T,rho,c,h,e decrease

Ma decreases:

• u decreases

• p,T,rho,c,h,e increase

Critical Ma Number * Critical Ma number after the shock = 1

No they dont need to be unique. Given an area you can have two solutions, one subsonic and one supersonic for example.

This kink appears at Ma = 1 (the critical pressure ratio). It occurs because at this point the flow has two different possible solutions. One would be the last possible subsonic case, the other the (isentropic) supersonic case

This comes from the enthalpy conservation and therefore doesnt require isentropy and also holds across shocks

Total:

• Total pressure decreases (entropy get generated -> total pressure loss)

• Total density decreases

• Total temperature stays the same (energy conservation)

• Total enthalpy stays the same (energy conservation)

• Total speed of sound stays the same

Static:

• Velocity decreases

• Ma number decreases

• Static pressure increases

• Static temperature increases

• Static density increases

• Static enthalpy increases

• Static speed of sound increases

  
  

Search for multiple solutions because of non linearity, then check whether these solutions are physically possible. The remaining ones are the normal shock solutions

The nozzle is choked (Ma = 1) in the smallest cross section. This means the mass flow density has its maximum and cannot be increased any further just by decreasing the nozzle back pressure (This can only be done via increasing the total density/pressure on the inlet or increasing the area)

Our theory is solely 1-D, some of the solutions however require more complex 2-D flow phenomena.

• Schlieren Photography

• Shadow imaging

• Mach Zehnder Intereferometry

Cuts out a portion of light at a given angle that can be set by the position of the knife. This cut out portion leads to darker regions in the image

Schlieren:

• First derivative of the density (direction can be controlled by the knife edge)

Shadow imaging:

• Second derivative of the density

Using schlieren imaging, you would position the knife in normal direction to the flow

Therefore you see the density gradient in wall normal direction

They some kind of direction and speed. If its hyperbolic in time, the speed is the real speed (propagation speed of information)

dphi = 0, if dx/dt = df(phi)/dphi.

-> They are constant because we build the characteristic curves with the characteristic velocity

1. Schlieren imaging: first derivative of density (direction depending on knife edge)

2. Shadowgraph imaging: second derivative of density

c*, doesnt change since it can be calculated through T01 or T02

Linear Processes:

• The shape of the characteristic quantity doesnt change

• Characteristic curves are parallel and cannot intersect

  -> No shocks in linear theory

Second derivative of the flux of the characteristic quantity is not zero (The characteristic velocity is not constant -> Non linear transport)

Convergent characteristics meet and intersect at a given timepoint -> Shock formation

(Information collides in a point, this causes a loss in information)

For a system of PDE’s, the characteristic velocities can be found as the eigenvalues of the system

(If all of the eigenvalues are real, its a hyperbolic system)

Linearized form:

Compression, Expansion/Rarefactions and Entropy waves

For Ma = 1, uref = cref

-> xdot = 0

No information transport through this characteristic

No, linear compression waves as their name suggests come from linear theory.

Shocks are non-linear and appear only in non-linear theory.

They have the same tendencies to increase pressure etc. however

“Linear compression waves are basically the limiting case when the shock intensity goes to 0”

It increases the velocity in running direction

-> The velocity becomes more negative (CoSy!)

It decreases the velocity in running direction

-> The velocity becomes more positive (CoSy!")

It decreases the velocity in running direction

-> The velocity becomes more negative (CoSy!)

It increases the velocity in running direction

-> The velocity becomes more positive (CoSy!)

1) Nonlinear smooth compression waves or non-smooth shock waves

2) Nonlinear expansion waves (probably as a fan, if centered)

3) Quasi-linear entropy waves

The nonlinearity makes the characteristic velocties dependent on the characteristic quantity itself.

-> The characteristic velocity is no longer constant -> They con-/diverge

No, shocks are nonlinear, linear compression waves are linear and sort of an edge case where the shock intensity goes to zero

Given a certain state of the gas before and after, there is a continuous! change of the state -> fan

No, they require the solution to be smooth and therefore not having any jumps!

If the fan is running to the left: the velocity increases

If the fan is running to the right: the velocity decreases

(Increases in opposite direction)

Yes:

Isentropic:

s = const

Homentropic:

s = const & grad(s) = 0

They are not the same

grad(h0) = 0

(For 2-D steady homentropic and isoenergetic flows with M>1)

No, the used quantities are only defined for Ma>=1

Expansions increase the relative angle

Compressions decrease the relative angle

In 2D, steady, inviscid, subsonic flow, lift can be generated but not drag. This contradicts reality -> paradox

Its a form of drag caused by waves, typically due to the losses in shockwaves. In linear theory there are no shockwaves but still the pressure imbalance in flow direction that causes the drag.

No, (shocks are always lossy)

No, thickness and camber cause drag, both of them arent contributing to lift in supersonic flight!

The only way to generate lift in supersonic flow, is through angulation

No. Since the only mechanism to create lift is through angle of attack this also causes wave drag

They are related through the tangent component, whether its existent or not.

-> If the tangent component of the oblique is zero it degenerates to a normal shock.

The Prandtl-Shock-Relation, shows that its not possible.

Yes, they can lead to supersonic, sonic and subsonic conditions

No, the direction of the expasion doesnt matter

For very weak shocks, the Ma number after the shock will be nearly the Ma Number before the shock

-> The Ma Angles are nearly the same -> Ma and Shock angle will be identical

Always to the local flow velocity!